Analytic in an unit ball functions of bounded L-index in joint variables

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Analytic in an unit ball functions

of bounded L-index in joint variables

Andriy Bandura, Oleh Skaskiv

(Presented by V. Ya. Gutlyanskii)

Abstract. A concept of boundedness of theL-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B.Suﬃcient condi-tions of boundedness of L-index in joint variables,Mat. Stud. 45(2016), 12–26. dx.doi.org/10.15330/ms.45.1.12-26) is generalized for analytic in a ball function. It is proved criteria of boundedness of theL-index in joint variables which describe local behavior of partial derivatives on a skeleton of a polydisc.

2010 MSC.32A05, 32A10, 32A30, 32A40, 30H99.

Key words and phrases.Analytic function, ball, boundedL-index in joint variables, maximum modulus, partial derivative.

1. Introduction

A concept of entire function of bounded index appeared in a paper of B. Lepson [23]. An entire function f is said to be of bounded index if there exists an integer N >0 that

(∀z∈C)(∀n∈ {0,1,2, . . .}) : |f (n)₍_z₎_| n! ≤max {_|_f(j)₍_z₎_| j! : 0≤j≤N } . (1.1)

The least such integerN is called the index of f.

Note that the functions from this class have interesting properties. The concept is convenient to study the properties of entire solutions of diﬀerential equations. In particular, if an entire solution has bounded index then it immediately yields its growth estimates, an uniform in a some sense distribution of its zeros, a certain regular behavior of the solution etc.

Received27.03.2017

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Afterwards, S. Shah [28] and W. Hayman [19] independently proved that every entire function of bounded index is a function of exponential type. Namely, its growth is at most the ﬁrst order and normal type.

To study more general entire functions, A. D. Kuzyk and M. M. She-remeta [21] introduced a boundedness of thel-index, replacing |f(p_p)_!(z)| on

|f(p)₍_z₎_|

p!lp₍_|_z_|₎ in(1.1), wherel:R+→R+is a continuous function. It allows to consider an arbitrary entire functionf with bounded multiplicity of zeros. Because for the functionf there exists a positive continuous functionl(z)

such that f is of bounded l-index [14]. Besides, there are papers where the deﬁnition of bounded l-index is generalizing for analytic function of one variable [22, 30].

In a multidimensional case a situation is more diﬃcult and interesting. Recently we with N. V. Petrechko [12, 13] proposed approach to consider bounded L-index in joint variables for analytic in a polydisc functions, where L(z) = (l1(z), . . . , ln(z)), lj : Cn → R+ is a positive continuous

functions,j∈ {1, . . . , n}.Although J. Gopala Krishna and S.M. Shah [20] introduced an analytic in a domain (a nonempty connected open set)

Ω⊂ Cn (n∈N) function of bounded index for α = (α1, . . . , αn) ∈ Rn+.

But analytic in a domain function of bounded index by Krishna and Shah is an entire function. It follows from necessary condition of the l-index boundedness for analytic in the unit disc function ( [29, Th.3.3,p.71]):

∫r

0 l(t)dt → ∞ as r → 1 (we take l(t) ≡ α1). Thus, there arises

neces-sity to introduce and to investigate bounded L-index in joint variables for analytic in polydisc domain functions. Above-mentioned paper [12] is devoted analytic in a polydisc functions. Besides a polydisc, other example of polydisc domain in Cn is a ball. There are two known mono-graphs [26, 32] about spaces of holomorphic functions in the unit ball of

Cn_: _{Bergman spaces, Hardy spaces, Besov spaces, Lipschitz spaces, the}

Bloch space, etc. It shows the relevance of research of properties of holo-morphic function in the unit ball. In this paper we will introduce and study analytic in a ball functions of boundedL-index in joint variables.

Of course, there are wide bibliography about entire functions of boun-ded L-index in joint variables [9–11, 15–18, 24, 25].

Note that there exists other approach to consider bounded index inCn

— so-called functions of bounded L-index in direction (see [1–7]), where L:Cn→R+ is a positive continuous function.

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2. Main definitions and notations

We need standard notations. Denote R+ = [0,+∞),0= (0, . . . ,0)∈ Rn +, 1 = (1, . . . ,1) ∈ R+n, 1j = (0, . . . ,0, |{z}1 j−th place ,0, . . . ,0) ∈ Rn₊, R = (r1, . . . , rn) ∈ Rn+, z = (z1, . . . , zn) ∈ Cn, |z| = √∑n j=1|zj|2.

For A = (a1, . . . , an) ∈ Rn, B = (b1, . . . , bn) ∈ Rn we will use

for-mal notations without violation of the existence of these expressions AB= (a1b1,· · ·, anbn), A/B = (a1/b1, . . . , an/bn), AB=ab11a

b2

2 ·. . .·abnn,

∥A∥ = a1 +· · ·+an, and the notation A < B means that aj < bj,

j ∈ {1, . . . , n}; the relation A ≤ B is deﬁned similarly. For K = (k1, . . . , kn) ∈ Zn+ denote K! = k1!·. . . ·kn!. Addition, scalar

multi-plication, and conjugation are deﬁned onCncomponentwise. For z∈Cn andw∈Cn we deﬁne

⟨z, w⟩=z1w1+· · ·+znwn,

wherewk is the complex conjugate of wk. The polydisc{z ∈Cn : |zj −

z_j0| < rj, j = 1, . . . , n} is denoted by Dn(z0, R), its skeleton {z ∈ Cn :

|zj −z0j| = rj, j = 1, . . . , n} is denoted by Tn(z0, R), and the closed

polydisc{z∈Cn: |zj−zj0| ≤rj, j = 1, . . . , n} is denoted byDn[z0, R],

Dn₌_Dn₍₀_,₁₎_,_D₌_{_z_∈_C_: _|_z_|_<₁_}_._{The open ball}_{_z_∈_Cn_: _|_z₋_z0_|_<

r}is denoted byBn(z0, r),its boundary is a sphereSn(z0, r) ={z∈Cn: |z−z0| = r}, the closed ball {z ∈ Cn : |z−z0| ≤ r} is denoted by

Bn_[_z0_{, r}_]_,_Bn₌_Bn₍₀_,₁₎_,_D₌_B1 ₌_{_z_∈_C_: _|_z_|_<₁_}_.

For K = (k1, . . . , kn)∈Zn+ and the partial derivatives of an analytic

inBn functionF(z) =F(z1, . . . , zn) we use the notation

F(K)(z) = ∂ ∥K∥_F ∂zK = ∂k1+···+kn_f ∂zk1 1 . . . ∂znkn .

Let L(z) = (l1(z), . . . , ln(z)), where lj(z) : Bn → R+ is a continuous

function such that

(∀z∈Bn) : lj(z)> β/(1− |z|), j ∈ {1, . . . , n}, (2.1)

whereβ >√n is a some constant.

S. N. Strochyk, M. M. Sheremeta, V. O. Kushnir [29, 30] imposed a similar condition for a functionl:D→R+ and l:G→R+, whereGis arbitrary domain inC.

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z∈Dn[z0, R/L(z0)] thenz∈Bn.Indeed, we have |z| ≤ |z−z0|+|z0| ≤ v u u t∑n j=1 r2 j l_j2(z0₎+|z 0_|_< v u u t∑n j=1 r2 j β2(1− |z 0_|₎2₊_|_z0_| = (1− |z 0_|) β v u u t∑n j=1 r2_j +|z0| ≤ (1− |z 0_|) β β+|z 0_|_{= 1}_.

An analytic function F:Bn→Cis said to be ofbounded L-index (in joint variables), if there exists n0 ∈Z+ such that for all z ∈Bn and for

all J ∈Zn₊ |F(J)(z)| J!LJ₍_z₎ ≤max { |F(K)(z)| K!LK₍_z₎ : K ∈Z n +, ∥K∥ ≤n0 } . (2.2)

The least such integer n0 is called the L-index in joint variables of the

functionF and is denoted byN(F,L,Bn) (see [9]– [16]).

By Q(Bn) we denote the class of functionsL, which satisfy (2.1)and the following condition

(∀R∈Rn₊,|R| ≤β, j∈ {1, . . . , n}) : 0< λ1,j(R)≤λ2,j(R)<∞, (2.3) where λ1,j(R) = inf z0_∈_Bninf { lj(z) lj(z0) :z∈Dn[z0, R/L(z0)]}, (2.4) λ2,j(R) = sup z0_∈_Bn sup { lj(z) lj(z0) :z∈Dn[z0, R/L(z0)]}. (2.5) Λ1(R) = (λ1,1(R), . . . , λ1,n(R)), Λ2(R) = (λ2,1(R), . . . , λ2,n(R)).

It is not diﬃcult to verify that the class Q(Bn) can be deﬁned as following: for every j∈{1, . . . , n}

sup z,w∈Bn { lj(z) lj(w) :|zk−wk| ≤ rk min{lk(z), lk(w)} , k∈ {1, . . . , n} } <∞, (2.6)

i. e. conditions(2.3)and (2.6)are equivalent.

Example 2.1. The function F(z) = exp{₍₁₋_z 1

1)(1−z2)} has bounded

L-index in joint variables with L(z) = (₍₁_−|_z 1

1|)2(1−|z|), 1 (1−|z|)(1−|z2|)2 ) and N(F,L,Bn) = 0.

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3. Local behavior of derivatives of function of bounded L-index in joint variables

The following theorem is basic in theory of functions of bounded in-dex. It was necessary to prove more eﬃcient criteria of index boundedness which describe a behavior of maximum modulus on a disc or a behavior of logarithmic derivative (see [1, 6, 22, 29]).

Theorem 3.1. Let L ∈ Q(Bn). An analytic in Bn function F has bounded L-index in joint variables if and only if for each R ∈ Rn₊,

|R| ≤ β, there exist n0 ∈ Z+, p0 > 0 such that for every z0 ∈ Bn

there existsK0 ∈Zn₊, ∥K0∥ ≤n0, and max { |F(K)(z)| K!LK₍_z₎:∥K∥ ≤n0, z∈D n[_z0_{, R/}_L(_z0₎] } ≤p0 | F(K0)(z0)| K0_!LK0 (z0₎. (3.1)

Proof. Let F be an analytic function of bounded L-index in joint vari-ables withN =N(F,L,Bn)<∞.For everyR ∈Rn₊,|R| ≤β,we put

q=q(R) = [2(N+ 1)∥R∥

n

∏

j=1

(λ1,j(R))−N(λ2,j(R))N+1] + 1

where [x] is the entire part of the real numberx,i.e. it is a ﬂoor function. Forp∈ {0, . . . , q} andz0∈Bn we denote

Sp(z0, R) = max { |F(K)(z)| K!LK₍_z₎ :∥K∥ ≤N, z∈D n [ z0, pR qL(z0₎ ]} , S_p∗(z0, R) = max { |F(K)(z)| K!LK₍_z0₎ :∥K∥ ≤N, z∈D n [ z0, pR qL(z0₎ ]} . Using (2.4) andDn [ z0,_q_LpR₍_z0₎ ] ⊂Dn[_z0_, R L(z0₎ ] , we have Sp(z0, R) = max { |F(K)(z)| K!LK₍_z₎ LK(z0) LK₍_z0₎ :∥K∥ ≤N, z∈D n [ z0, pR qL(z0₎ ]} ≤ S_p∗(z0, R) max    n ∏ j=1 lN_j (z0) lN j (z) :z∈Dn [ z0, pR qL(z0₎ ]_  ≤ S_p∗(z0, R) n ∏ j=1 (λ1,j(R))−N.

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and, using (2.5), we obtain S_p∗(z0, R) = max { |F(K)(z)| K!LK₍_z₎ LK(z) LK₍_z0₎ :∥K∥≤N, z∈D n [ z0, pR qL(z0₎ ]} ≤ max { |F(K)(z)| K!LK₍_z₎(Λ2(R)) K _:_∥_K_{∥ ≤}_{N, z}_∈_Dn [ z0, pR qL(z0₎ ]} ≤ Sp(z0, R) n ∏ j=1 (λ2,j(R))N. (3.2)

LetK(p)∈Zn₊ with ∥K(p)∥ ≤N and z(p)∈Dn

[ z0,_q_LpR₍_z0₎ ] be such that S_p∗(z0, R) = |F (K(p))₍_z(p)₎_| K(p)_!LK(p) (z0₎ (3.3)

Since by the maximum principle z(p)∈Tn(z0,_q_LpR₍_z0₎),we havez(p)̸=z0.

We choose ze_j(p) =z0_j +p−_p1(z_j(p)−z_j0).Then for every j ∈ {1, . . . , n} we have |ze(_jp)−z_j0| = p−1 p |z (p) j −z 0 j|= p−1 p prj qlj(z0) , (3.4) |ze_j(p)−z_j(p)| = |z_j0+p−1 p (z (p) j −z 0 j)−z (p) j |= 1 p|z 0 j −z (p) j | = 1 p prj qlj(z0) = rj qlj(z0) . (3.5) From (3.4) we obtainze(p)∈Dn [ z0,_q₍(_Rp−₎_L1)₍_zR0₎ ] and S_p∗₋₁(z0, R)≥ |F (K(p)₎ (ze(p))| K(p)_!LK(p) (z0₎. From (3.3) it follows that

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= 1 K(p)_!LK(p) (z0₎ × n ∑ j=1 |z_j(p)−ze_j(p)|F(K(p)+1j)₍e_z(p)₊_t∗₍_z(p)−e_z(p)₎₎_, _(3.6) where 0 ≤ t∗ ≤ 1,ez(p) +t∗(z(p) −ze(p)) ∈ Dn(z0,_q_LpR₍_z0₎). For z ∈ Dn₍_z0_, pR qL(z0₎) and J ∈Zn+,∥J∥ ≤N+ 1 we have |F(J)₍_z₎_|_LJ₍_z₎ J!LJ₍_z0_)LJ₍_z₎ ≤(Λ2(R)) J_max { |F(K)₍_z₎_| K!LK₍_z₎ :∥K∥ ≤N } ≤ n ∏ j=1 (λ2,j(R))N+1(λ1,j(R))−Nmax { |F(K)(z)| K!LK₍_z0₎ :∥K∥ ≤N } ≤ n ∏ j=1 (λ2,j(R))N+1(λ1,j(R))−NSp∗(z0, R).

From (3.6) and (3.5) we obtain 0 ≤ S_p∗(z0, R)−S_p∗₋₁(z0, R)≤ n ∏ j=1 (λ2,j(R))N+1(λ1,j(R))−N × S_p∗(z0, R) n ∑ j=1 (k_j(p)+ 1)lj(z0)|zj(p)−ez (p) j | = n ∏ j=1 (λ2,j(R))N+1(λ1,j(R))−N S_p∗(z0, R) q(R) n ∑ j=1 (k_j(p)+ 1)rj ≤ n ∏ j=1 (λ2,j(R))N+1(λ1,j(R))−N S_p∗(z0, R) q(R) (N + 1)∥R∥ ≤ 1 2S ∗ p(z0, R).

This inequality impliesS_p∗(z0, R)≤2S_p∗₋₁(z0, R),and in view of inequa-lity (3.2) we have Sp(z0, R)≤2 n ∏ j=1 (λ1,j(R))−NSp∗−1(z0, R) ≤2 n ∏ j=1 (λ1,j(R))−N(λ2,j(R))NSp−1(z0, R) Therefore, max { |F(K)(z)| K!LK₍_z₎ :∥K∥ ≤N, z∈D n [ z0, pR qL(z0₎ ]} =Sq(z0, R)

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≤ 2 n ∏ j=1 (λ1,j(R))−N(λ2,j(R))NSq−1(z0, R)≤. . . ≤ (2 n ∏ j=1 (λ1,j(R))−N(λ2,j(R))N)qS0(z0, R) = (2 n ∏ j=1 (λ1,j(R))−N(λ2,j(R))N)q × max { |F(K)(z0)| K!LK₍_z0₎ :∥K∥ ≤N } . (3.7)

From (3.7) we obtain inequality (3.1) with

p0= (2

n

∏

j=1

(λ1,j(R))−N(λ2,j(R))N)q

and someK0with∥K0∥ ≤N. The necessity of condition (3.1) is proved. Now we prove the suﬃciency. Suppose that for every R∈Rn

+,|R| ≤

β, there exist n0 ∈ Z+, p0 > 1 such that for all z0 ∈ Bn and some

K0 ∈Zn₊,∥K0∥ ≤n0,the inequality (3.1) holds.

We write Cauchy’s formula as following ∀z0_∈_Bn _∀_k_∈_Zn

+ ∀S∈Zn+ F(K+S)(z0) S! = 1 (2πi)n ∫ Tn(_z0_, R L(z0) ) F (K)₍_z₎ (z−z0₎S+1dz.

Therefore, applying (3.1), we have |F(K+S)(z0)| S! ≤ 1 (2π)n ∫ Tn(_z0_, R L(z0) ) |F (K)₍_z₎_| |z−z0|S+1|dz| ≤ ∫ Tn(_z0_, R L(z0) )|F(K)(z)| L S+1₍_z0₎ (2π)n_RS+1|dz| ≤ ∫ Tn(_z0_, R L(z0) )|F(K0)(z0)| × K!p0 ∏n j=1λ n0 2,j(R)LS+K+1(z0) (2π)n_K0_!_RS+1_LK0 (z0₎ |dz| = |F(K0)(z0)|K!p0 ∏n j=1λ n0 2,j(R)LS+K(z0) K0_!_RS_LK0 (z0₎ . This implies |F(K+S)(z0)| (K+S)!LS+K₍_z0₎ ≤ ∏n j=1λ n0 2,j(R)p0K!S! (K+S)!RS |F(K0)(z0)| K0_!LK0 (z0₎. (3.8)

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Obviously, that K!S! (K+S)! = s1! (k1+ 1)·. . .·(k1+s1)· · · sn! (kn+ 1)·. . .·(kn+sn) ≤ 1. We choose rj ∈ (1, β/√n], j ∈ {1, . . . , n}. Then |R| = √∑n_j₌₁r_j2 ≤ β. Hence, p0 ∏n j=1λ n₀ 2,j(R)

RS →0 as ∥S∥ →+∞.Thus, there existss0 such that

for all S∈Zn₊ with∥S∥ ≥s0 the inequality holds

p0K!S! ∏n j=1λ n0 2,j(R) (K+S)!RS ≤1. Inequality (3.8) yields ₍_K|₊F_S(K_)!+_LSK)(+zS0)₍_z|0₎ ≤ |F(K0)₍_z0₎_|

K0_!_LK0₍_z0₎.This means that for

everyj∈Zn₊ |F(J)(z0)| J!LJ₍_z0₎ ≤max { |F(K)(z0)| K!LK₍_z0₎ :∥K∥ ≤s0+n0 }

where s0 and n0 are independent of z0. Therefore, the analytic in Bn function F has bounded L-index in joint variables with N(F,L,Bn) ≤

s0+n0.

Theorem 3.2. Let L∈Q(Bn).In order that an analytic inBn function F be of bounded L-index in joint variables it is necessary that for every R ∈ Rn +, |R| ≤ β, ∃n0 ∈ Z+ ∃p ≥ 1 ∀z0 ∈ Bn ∃K0 ∈Z+n, ∥K0∥ ≤ n0, and max { |F(K0)(z)|:z∈Dn[z0, R/L(z0)]}≤p|F(K0)(z0)| (3.9)

and it is suﬃcient that for every R ∈ Rn₊, |R| ≤ β, ∃n0 ∈ Z+ ∃p ≥ 1 ∀z0 _∈ _Bn _∀_j _{∈ {}₁_{, . . . , n}_{} ∃}_K0 j = (0, . . . ,0, _|{z}kj0 j-th place ,0, . . . ,0) such that k_j0≤n0 and max { |F(Kj0)₍_z₎|_:_z∈Dn[_z0_{, R/}_L(_z0₎]}≤_p|_F(Kj0)₍_z0₎|_. _(3.10) Proof. Proof of Theorem 3.1 implies that the inequality (3.1) is true for someK0.Therefore, we have

p0 K0_! |F(K0)(z0)| LK0 (z0₎ ≥max { |F(K0)(z)| K0_!LK0 (z) :z∈D n[_z0_{, R/}_L(_z0₎] }

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= max { |F(K0)(z)| K0_! LK0(z0) LK0 (z0_)LK0 (z) :z∈D n[_z0_{, R/}_L(_z0₎] } ≥ max { |F(K0)(z)| K0_! ∏n j=1(λ2,j(R))− n0 LK0 (z0₎ :z∈D n[_z0_{, R/}_L(_z0₎] } .

This inequality implies

p0 ∏n j=1(λ2,j(R))n0 K0_! |F(K0)(z0)| LK0 (z0₎ ≥ max { |F(K0)(z)| K0_!LK0 (z0₎ :z∈D n[_z0_{, R/}_L(_z0₎] } . (3.11)

From (3.11) we obtain inequality (3.9) withp=p0 ∏n

j=1(λ2,j(R))

n0_{. The}

necessity of condition (3.9) is proved.

Now we prove the suﬃciency of (3.10). Suppose that for every R ∈

Rn

+, |R| ≤ β, ∃n0 ∈ Z+, p > 1 such that ∀z0 ∈ Bn and some KJ0 ∈ Zn+ withk0_j ≤n0 the inequality (3.10) holds.

We write Cauchy’s formula as following ∀z0∈Bn ∀S∈Zn₊ F(KJ0+S)(z0) S! = 1 (2πi)n ∫ Tn₍_z0_,R/_L₍_z0₎₎ F(KJ0)(z) (z−z0₎S+1dz. This yields |F(Kj0+S)₍_z0)| S! ≤ 1 (2π)n ∫ Tn₍_z0_,R/_L₍_z0₎₎ |F(Kj0)₍_z)| |z−z0_|S+1|dz| ≤ 1 (2π)nmax{|F (K0 j)₍_z)|_:_z∈Dn[_z0_{, R/}_L(_z0₎]}L S+1₍_z0₎ RS+1 × ∫ Tn₍_z0_,R/_L₍_z0₎₎| dz| = max{|F(Kj0)₍_z₎|_:_z∈Dn[_z0_{, R/}_L(_z0₎]}L S₍_z0₎ RS . Now we putR = (√β n, . . . , β √ n) and use (3.10) |F(Kj0+S)₍_z0₎| S! ≤ LS₍_z0₎ (β/√n)∥S∥max{|F (K0 j)₍_z)|_:_z∈Dn[_z0_{, R/}_L(_z0₎]} ≤ pLS(z0) (β/√n)∥S∥|F (K0 j)₍_z0₎|_. _(3.12)

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We choose S ∈Zn₊ such that ∥S∥ ≥ s0, where ₍_β/√p_n₎s₀ ≤1. Therefore,

(3.12) implies that for allj∈ {1, . . . , n} and k_j0≤n0 |F(Kj0+S)₍_z0₎| LKj0+S₍_z0₎₍_K0 j+S)! ≤ p (β/√n)∥S∥ S!K_j0! (S+K0 j)! |F(Kj0)₍_z0₎| LKj0₍_z0₎_K0 j! ≤ |F (K0 j)₍_z0₎| LKj0₍_z0₎_K0 j! . Consequently,N(F,L,Bn)≤n0+s0.

Remark 3.1. Inequality (3.9) is necessary and sufficient condition of boundedness of l-index for functions of one variable [22, 29, 31]. But it is unknown whether this condition is sufficient condition of boundedness of L-index in joint variables. Our restrictions (3.10) are corresponding multidimensional sufficient conditions.

Lemma 3.1. Let L1,L2 ∈Q(Bn) and for every z∈Bn L1(z) ≤L2(z).

If analytic in Bn function F has bounded L1-index in joint variables then F is of bounded L2-index in joint variables and N(F,L2,Bn) ≤

nN(F,L1,Bn).

Proof. LetN(F,L1,Bn) =n0.Using (2.2) we deduce |F(J)(z)| J!LJ₂(z) = LJ₁(z) LJ₂(z) |F(J)(z)| J!LJ₁(z) ≤ LJ1(z) LJ 2(z) max { |F(K)(z)| K!LK 1 (z) : K∈Zn₊, ∥K∥ ≤n0 } ≤ LJ1(z) LJ 2(z) max { LK₂ (z) LK 1 (z) |F(K)(z)| K!LK 2 (z) : K ∈Zn₊, ∥K∥ ≤n0 } ≤ max ∥K∥≤n0 ( L1(z) L2(z) )J−K max { |F(K)(z)| K!LK 2 (z) : K ∈Zn₊, ∥K∥ ≤n0 } .

Since L1(z)≤L2(z) it means that for all∥J∥ ≥nn0 |F(J)(z)| J!LJ 2(z) ≤max { |F(K)(z)| K!LK 2 (z) : K∈Zn₊, ∥K∥ ≤n0 } .

Thus, F has bounded L2-index in joint variables and N(F,L2,Bn) ≤

nN(F,L1,Bn).

Denote L(e z) = (el1(z), . . . ,eln(z)). The notation L ≍ Le means that

there exist Θ1 = (θ1,j, . . . , θ1,n) ∈ Rn+, Θ2 = (θ2,j, . . . , θ2,n) ∈ Rn+ such

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Theorem 3.3. Let L∈Q(Bn), L≍Le, β|Θ1|>√n. An analytic in Bn function F has bounded Le-index in joint variables if and only if it has bounded L-index.

Proof. It is easy to prove that ifL∈Q(Bn) andL≍Le thenLe∈Q(Bn).

Let N(F,Le,Bn_{) =} _e_n

0 < +∞. Then by Theorem 3.1 for every Re = (re1, . . . ,ren)∈Rn+,|R| ≤β,there exists pe≥1 such that for each z0 ∈Bn and someK0 with ∥K0∥ ≤ne0, the inequality (3.1) holds with Le and Re instead ofL and R. Hence

e p K0_! |F(K0)(z0)| LK0 (z0₎ = e p K0_! ΘK₂ 0|F(K0)(z0)| ΘK₂ 0LK0 (z0₎ ≥ e p K0_! |F(K0)(z0)| ΘK0 2 LeK 0 (z0₎ ≥ 1 ΘK0 2 max { |F(K)₍_z₎_| K!LeK₍_z₎ :∥K∥ ≤en0, z∈D n[_z0_,_R/e _L(e _z₎]} ≥ 1 ΘK₂0 max { ΘK 1 |F(K)(z)| K!LK₍_z₎ :∥K∥ ≤en0, z∈D n[_z0_,_Θ1_R/e _L(_z₎]} ≥ min0≤∥K∥≤n0{Θ K 1 } ΘK₂0 × max { |F(K)(z)| K!LK₍_z₎ :∥K∥ ≤ne0, z∈D n[_z0_,_Θ 1R/e L(e z) ]} .

In view of Theorem 3.1 we obtain that functionF has bounded L-index in joint variables.

Theorem 3.4. Let L ∈ Q(Bn). An analytic in Bn function F has bounded L-index in joint variables if and only if there exist R ∈ Rn₊, with |R| ≤β, n0 ∈Z+, p0 >1 such that for eachz0 ∈Bn and for some

K0 ∈Zn₊ with∥K0∥ ≤n0 the inequality (3.1)holds.

Proof. The necessity of this theorem follows from the necessity of The-orem 3.1. We prove the suﬃciency. The proof of TheThe-orem 3.1 with

R= (√β_n, . . . ,√β_n) implies that N(F,L,Bn)<+∞.

LetL∗(z) = R0L_R(z), R0 _{= (}_√β n, . . . ,

β

√

n).In general case from validity

of (3.1) forF,L and R= (r1, . . . , rn) with|R| ≤β, R̸=R0,we obtain

max { |F(K)(z)| K!(L∗(z0₎₎K :∥K∥ ≤n0, z∈D n[_z0_{, R} 0/L∗(z0) ]} = max { |F(K)(z)| K!(R0L(z)/R)K :∥K∥ ≤n0, z∈Dn [ z0, R0/(R0L(z)/R) ]}

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≤ max { n∥K∥/2|F(K)(z)| K!LK₍_z₎ :∥K∥ ≤n0, z∈D n[_z0_{, R/}_L(_z0₎] } ≤ p0 K0_! nn0/2|_F(K0)₍_z0₎| LK0 (z0₎ = nn0/2₍_β/√_n₎∥K0∥_p 0 RK0 K0_! |F(K0)(z0)| (R0L(z)/R)K0 < nn0/2_p 0 max ∥K0_∥≤_n 0 (β/√n)∥K0∥ RK0 |F(K0)(z0)| K0_!(L∗₍_z₎₎K0.

i. e. (3.1) holds for F, L∗ and R0 = (√β_n, . . . ,√β_n).As above we apply Theorem 3.1 to the functionF(z) and L∗(z) =R0L(z)/R. This implies thatF is of bounded L∗-index in joint variables. Therefore, by Theorem 3.3 the functionF has boundedL-index in joint variables.

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Contact information Andriy Bandura Ivano-Frankivsk National

Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine

E-Mail: andriykopanytsia@gmail.com

Oleh Skaskiv Ivan Franko National University of Lviv, Lviv, Ukraine